For decision making under uncertainty, a rational optimality criterion is min-max. Min-max problems such that the minimizer makes an optimal decision against the worst case that might be chosen by the maximizer are studied. This paper presents necessary conditions and computational methods for a min-max solution (not a saddle point solution). Those conditions are stated in a form like Kuhn-Tucker theorem. The computational methods are based on the relaxation procedure. A min-max problem such that the minimizer and the maximizer are subject to separate constraints is primarily studied. But it is shown that the obtained results can be applied for the unseparate constraint case by use of duality theory.

Necessary conditions for min-max problems and algorithms by a relaxation procedure

This paper is concerned with the Stackelberg problem and the min-max problem in competitive systems. The Stackelberg approach is applied to the optimization of two-level systems where the higher level determines the optimal value of its decision variables (parameters for the lower level) so as to minimize its objective, while the lower level minimizes its own objective with respect to the lower level decision variables under the given parameters. Meanwhile, the min-max problem is to determine a min-max solution such that a function maximized with respect to the maximizer's variables is minimized with respect to the minimizer's variables. This problem is also characterized by a parametric approach in a two-level scheme. New computational methods are proposed here; that is, a series of nonlinear programming problems approximating the original two-level problem by application of a penalty method to a constrained parametric problem in the lower level are solved iteratively. It is proved that a sequence of approximated solutions converges to the correct Stackelberg solution, or the min-max solution. Some numerical examples are presented to illustrate the algorithms.

A new computational method for Stackelberg and min-max problems by use of a penalty method

In this paper, we investigate the design of pulse shaping UK filters for impulse radio ultrawideband (UWB) communications systems. The goal of the shaping is to meet an arbitrary spectrum mask, e.g., the mask mandated by the FCC for UWB emissions. Compared with classical FIR filter designs, the current problem introduces three new challenges: (1) it is minimax with quadratic constraints, (2) a single-sided distortion function is used, (3) delay positions are treated as tuning parameters. We first approach this problem by constructing a least-squares approximation to the minimax problem where the optimization over delays can be easily solved. With the LS solution serving as an initialization, nonlinear optimization techniques are employed to fine tune the solutions.

Impulse radio pulse shaping for ultra-wide bandwidth (UWB) systems

Communication satellites have evolved from a radio relay in space to encompass other applications such as processing and meteorology. The demand for high capacity communication satellites has led in the past two decades to the development of multibeam antennas, a landmark in satellite antenna technology. Satellite antennas are classified into three categories according to function: communication; tracking, telemetry, and command (TT and C); and special satellite antennas. While the application of shaped beam antennas has been developed to its fullest extent as in INTELSAT VI and VII, an active monolithic microwave integrated circuit (MMIC) phased array will play an important role in future satellite antenna technology. Examples of each state-of-the-art satellite antenna technology and the process of space qualification are presented briefly. >

Satellite antennas

An essential part of realizing a shaped beam by using a multiple-beam antenna is determining the excitation coefficient for each individual port. An algorithm which optimizes the power gain in a minimax sense is described. This criterion, although not as mathematically tractable as least squares, is necessary when the specifications prescribe that the worst case be as good as possible. It will be shown how the basic algorithm is very efficient and how it permits simple modifications to allow the specification of one-sided errors for the pattern, and direct optimization of gain slopes. A design of a C-band shaped-beam satellite antenna provides an example.

Design of shaped-beam antennas through minimax gain optimization

An algorithm for minimax parameter optimization

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An algorithm for minimax parameter optimization